Willow song (traditional english c. 1600), Chaucer Fielder

This curious and intriguing old song has come down to us because it features in one of William Shakespeare's plays, Othello. I found it not while looking at the play, but in a 1950s collection of songs featuring in Shakespeare's work compiled by H.A. Chambers, giving the melodies and with piano arrangements. Found in one of my local charity shops.
Most of the songs he catalogues were set to music by classical composers, from Thomas Arne and Henry Purcell in the seventeenth century to Schubert and Mendelssohn in the nineteenth. But there are three which are traditional folk songs, including this one.
It feels to me like this song has a very unsettled character, changing between keys and also between minor and major, which paradoxically make it feel quite modern.
I have looked it up online, and most youtube renditions involve people in period costumes accompanied on lutes!! This is a step too far for me as I don't have the right kit, instrument or motivation!! So I've drawn upon some folky guitar ideas that I know of, to see what happens, this is the experimental result